Skip to main content

Full text of "Mathematical And Physical Papers - Iii"

Taking polar coordinates r, 0 in the plane of f', 17', we have
(sin 0 + co cos 0) (cos 9dr- sin 0 rd0)
= (cos 0  o) sin 0) (sin 8 dr + cos 0 rcZ0);
whence                                codr^rdO ........................ (24),
the differential equation of a system of equiangular spirals in which the angle between the tangent and radius vector is equal to tan""1^. We have also from (23)
a    i        2\-l   ) b  ^"s     '   '/ ^'/      I          b  w'/ + &)     }----<^->----75------T^ -----f.7T
I                         whence              (1 4- w2) log f' + const = log r + 0
, j                         = log r + &>2 log r from (24).    We have therefore
? = mr...........................(25),
where m is an arbitrary constant.
1                               Hence, in the plane of ', 77', conceive an equiangular spiral
j!   ;                          described about the origin as pole, such that the angle between
|    '                         the tangent and the radius vector is equal to tan^o).    Let it
:                            assume all  possible positions by turning once round the pole,
  '                          and in each position let it be made the base of a cylinder whose
j                             generating lines are parallel to the axis of '.    Conceive also an
I   t[                          infinite number of cones of revolution described with the origin !  '                           for vertex and the axis of ' for axis.    The system of curves of
M                             double curvature formed by the intersection of the cones with
II  r                                            ^                                     y
$ \'                          the cylinders will be the lines of motion in the auxiliary solid,
||                             on the  supposition  that  the  constant o> does  not vanish.    To
j;                             obtain the lines of motion in the original solid, it will be suf-
;!                             ficient to conceive the whole figure differently magnified or di-
i                             minished in three rectangular directions, and we shall thus obtain
a clearer idea of the form of the curves, which is the whole object
of the investigation, than would have been derived from the rather
!                             complicated equations got from the integration of equations (20)
in their original shape.
21. It may be observed, in conclusion, that even if there were reason to suppose that the constants DI} EI} FI} were not necessarily equal to zero, it is only among crystals which possess